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The Product of Automorphic Weighted Composition Operators on Hardy Space H <sup>2</sup>
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Abstract<p>Let <inline-formula> <tex-math><?CDATA $n\in {\mathbb{N}},{p}_{i}\in {\rm{U}},{\alpha }_{{P}_{i}}(z)=\frac{{p}_{i}-z}{1-{\bar{p}}_{i}z}(z\in {\rm{U}})$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <mi>n</mi> <mo>∈</mo> <mi>ℕ</mi> <mo>,</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi mathvariant="normal">U</mi> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>−</mo> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mover accent="true"> <mi>p</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> <mi>z</mi> </mrow> </mfrac> <mo stretchy="false">(</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="normal">U</mi> <mo stretchy="false">)</mo> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn1.gif" xlink:type="simple"></inline-graphic> </inline-formula>, and let <inline-formula> <tex-math><?CDATA ${f}_{1}\in {H}^{\infty },i=1,\ldots,n$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>∈</mo> <msup> <mi>H</mi> <mi>∞</mi> </msup> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn2.gif" xlink:type="simple"></inline-graphic> </inline-formula>. We discuss the relation between the points <italic>p<sub>i</sub> </italic> in U and the functions <italic>f<sub>i</sub> </italic> in U and the properties of the product of automorphic weighted composition operators <inline-formula> <tex-math><?CDATA ${W}_{{f}_{1},{\alpha }_{{p}_{1}}}\,{W}_{{f}_{2},{\alpha }_{{p}_{2}}}\ldots {W}_{{f}_{i},{\alpha }_{pi}}$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </msub> <mspace width="0.25em"></mspace> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </msub> <mo>…</mo> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn3.gif" xlink:type="simple"></inline-graphic> </inline-formula> on Hardy space H<sup>2</sup>. In fact, it is very nice connection between analytic function theory and operator theory. In this paper, we give the sufficient and necessary conditions to be normal, unitary, hermitian operator on <italic>H</italic> <sup>2</sup> and we shall present the shape of the numerical range of it.</p>
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Publication Date
Sat Jun 30 2007
Journal Name
Iraqi Journal Of Chemical And Petroleum Engineering
Anodic Polarization of Mild Steel in Saturated C a(OH)2 Contaminated with NaCl in Presence of NaNO2
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The corrosion  behavior  of mild sleet  in saturated  aerated  and de-aerated  Ca(OH)2 solution  was  investigated  using electrochemical measurements. The work was carried out with small coupons immersed in solutions containing different quantities of NaCl in presence of various NaN02 concentrations as corrosion inhibitors. It has been found thal:(1 ) In presence  of NaCl,  the time required  to reach O2  evolution potential  in de-aerated  Ca(OH)2 polarized  at  10μA/cm 2 is function of inhibitor concentration and it becomes lass as NaN02 increases compared with zero presence indicating the effectiveness of N

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Publication Date
Fri Mar 01 2024
Journal Name
Baghdad Science Journal
Synthesis and Characterization of New 2-amino-5-chlorobenzothiazole Derivatives Containing Different Types of Heterocyclic as Antifungal Activity
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Nine new compounds of 2-amino-5-chlorobenzothiazole derivatives were synthesized. These new compounds were formed through the reaction of 2-amino-5-chlorobenzothiazole 1 with ethyl chloroacetate and KOH, which gave an ester derivative 2, followed by refluxing compound 2 with hydrazine hydrate to afford hydrazide derivative 3. The reaction of compound 3 with CS2 and KOH gave 1,3,4-oxadiazole-2-thiol derivative 4, and then the reaction of compound 2 with thiosemicarbazide to produce compound 5  then treated it with 4%NaOH led to ring closure to provide 1,2,4-triazole-3-thiol derivative

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Publication Date
Tue Jan 01 2019
Journal Name
Baghdad Science Journal
Synthesis of new five-membered hetrocyclic compounds from 2-furfuryl mercaptan derivative and evaluation of their biological activity
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A theoretical and protection study was conducted of the corrosion behavior of carbon steel surface with different concentrations of the derivative (Quinolin-2-one), namly (1-Amino-4,7-dimethyl-6-nitro-1H-quinolin-2-one (ADNQ2O)). Theoretically, Density Functional Theory (DFT) of B3LYP/ 6-311++G (2d, 2p) level was used to calculate the optimized geometry, physical properties and chemical inhibition parameters, with the local reactivity to predict both the reactive centers and to locate the possible sites of nucleophilic and electrophilic attacks, in vacuum, and in two solvents (DMSO and H2O), all at the equilibrium geometry. Experimentally, the inhibition efficiencies (%IE) in the saline solution (of 3.5%) NaCl were studied using potentiomet

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Scopus
Publication Date
Thu Aug 31 2017
Journal Name
Journal Of Engineering
Study and Analysis of Concentric Shell and Double Tube Heat Exchanger Using - Al 2 O 3 Nanofluid
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    Heat exchanger is an important device in the industry for cooling or heating process. To increase the efficiency of heat exchanger, nanofluids are used to enhance the convective heat . transfer relative to the base fluid. - Al2O3/water nanofluid is used as cold stream in the shell and double concentric tube heat exchanger counter current to the hot stream basis oil. These nanoparticles were of particle size of 40 nm and it was mixed with a base fluid (water) at volume
concentrations of 0.002% and 0.004%. The results showed that each of Nusselt number and overall heat transfer coefficient increased as nanofluid concentrations increased. The pressure drop of nanofluid increased slightly than the base fluid because

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Publication Date
Sat Sep 11 2010
Journal Name
Journal Of Al-nahrain University
ESTIMATION ACTIVITY OF LAP IN PATIENT S WITH TYPE 2 DIABETES BY USING LEUCINE AMIDE AS SUBSTRATE
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This study was performd on 50 serum specimens of patients with type 2 diabetes, in addition, 50 normal specimens were investigated as control group. The activity rate of LAP in patients (560.46 10.504) I.U/L and activity rate of LAP in healthy(10.58 4.39)I.U/L.The results of the study reveal that Leucine aminopeptidase (LAP) activity of type 2 diabetes patient s serum shows a high signifiacant increase (p < 0.001) compare to healthy subjects. Addition preparation leucine amide as substrate of LAP, identification melting point and spectra by FTIR. K

Publication Date
Sun Jun 05 2016
Journal Name
Baghdad Science Journal
Complexes of Some Transition Metal with 2-Benzoyl thiobenzimidazole and 1,10-Phenanthroline and Studying their Antibacterial Activity
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Mixed ligands of 2-benzoyl Thiobenzimiazole (L1) with 1,10-phenanthroline (L2) complexes of Cr(III) , Ni(II) and Cu(II) ions were prepared. The ligand and the complexes were isolated and characterized in solid state by using FT-IR, UV-Vis spectroscopy, 1H, 13C-NMR, flame atomic absorption, elemental micro analysis C.H.N.S, magnetic susceptibility , melting points and conductivity measurements. 2-Benzoyl thiobenzimiazole behaves as bidenetate through oxygen atom of carbonyl group and nitrogen atom of imine group. From the analyses Octahedral geometry was suggested for all prepared complexes. A theoretical treatment of ligands and their metal complexes in gas phase were studied using HyperChem-8 program, moreover, ligands in gas phase

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Crossref
Publication Date
Sun Dec 01 2019
Journal Name
Journal Of Medicinal And Chemical Sciences
Synthesis and Identification of New 2-Substituted-1,3,4-Oxadiazole Compounds from Creatinine and Study Their Antioxidant Activities
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Publication Date
Sun Jul 01 2018
Journal Name
Journal Of Global Pharma Technology
Synthesis of New 2, 4, 5-triphenyl imidazole Derivatives Derived from benzoin and Studying their Biological Activity
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compound [1] was formed from the reaction of benzoin and benzaldehyde in the presence of ammonia, which was reacted with sodium hydride in DMF to obtain imidazole salt. This salt was reacted with adipoyl chloride to give compound [2]. Acid hydrazide derivative [3] was obtained from the reaction of compound [2] with hydrazine hydrate. After that Shiff bases [4-9] have been synthesized from the reaction of compound [3] with different aromatic aldehydes. These new formed compounds were diagnosed by 13C-NMR, 1H-NMR for some of them (in Ahl-Albate University in Jordan) and FT-IR spectroscopy (In Baghdad University). All of the prepared products have been studied their biological activities toward two kinds of bacteria. These products show

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Scopus
Publication Date
Wed Feb 01 2023
Journal Name
Baghdad Science Journal
Evaluation of Beta-2 Integrin and Platelets Roles in Sickle Cell Disease Pathogenicity in Basrah Governorate Patients
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Sickle cell disease (SCD) comprises an inherited blood disorder that is life long and affects many people globally. In spite of the development in treatment, SCA is a considerable cause of mortality and morbidity. The present study tries to assess the role of leukocytes represented by β integrin(CD18) and platelets and their productivity in the pathogenicity of disease during  the steady state and crisis in comparison with the healthy as-control group, SCD patients (15) enrolled during crisis and steady state (follow up) showed a significant increase in leukocytes and platelets cells productivity during crisis when compared to the steady state and in the steady state when compared to the healthy control group . In this study, SCD patho

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Publication Date
Wed Mar 29 2017
Journal Name
Iraqi Journal Of Pharmaceutical Sciences ( P-issn 1683 - 3597 E-issn 2521 - 3512)
Design and Synthesis of New Non-Steroidal Anti-inflammatory Agents with Expected Selectivity toward Cyclooxygenase-2 Inhibition
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This study includes design and synthesis of new non-steroidal anti-inflammatory agents (NSAIDs) with expected cyclooxygenase-2 (COX-2) selective inhibition to achieve better activity and low gastric side effects. Two series of compounds have been designed and synthesized as potential NSAIDs,these  are:     Salicylamide derivatives (compounds 3,4,5 ) and Diflunisal derivatives (compounds 10&11). In vivo acute anti-inflammatory effect of one of the synthesized agents (compound 3)  was evaluated in the rat using egg-white induced paw edema model of inflammation. Preliminary pharmacological study revealed that compound 3 exhibited less anti-inflammatory effect  compared to that of aspirin after

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